Download The Game of Logic by Habib Bin Muzaffar

The Game of Logic
by
Habib Bin Muzaffar
Dictionary Definiton of Logic
(Merriam-Webster Dictionary)
1. A proper or reasonable way of thinking about
or understanding something
2. A particular way of thinking about something
3. The science that studies the formal process
used in thinking and reasoning
The beginning of Mathematics
• Should one start with the first methodical deductions in
geometry traditionally credited to Thales of Miletus
around 600 B. C.? Or should one go back further and
start with the empirical derivation of certain
mensuration formulas made by the pre-Greek
civilizations of Mesopotamia and Egypt? Or should one
go back even further and start with the first groping
efforts made by prehistoric man to systematize size,
shape and number?........Or was mathematics, as Plato
believed, always in existence, merely awaiting
discovery? ( An Introduction to the History of
Mathematics by Howard Eves)
Aristotle’s statement in his
Metaphysics
• When all the inventions had been discovered,
the sciences which are not concerned with the
pleasures and necessities of life were
developed first in the lands where men began
to have leisure. This is the reason why
mathematics originated in Egypt, for there the
priestly class was able to enjoy leisure. ( The
History of Mathematics, An Introduction by
David M. Burton)
Thales of Miletus (circa 625-547 B. C)
1. First known person with whom specific
mathematical discoveries are traditionally
associated
2. Credited with introducing proofs in
mathematics
3. Credited with lots of wise statements
4. When asked what was the strangest thing he
had ever seen, he is said to have answered “
An aged tyrant”
The Flourishing of Greek Mathematics
(600-200 B.C.)
1. Extensive discoveries especially in geometry
2. Discovery of irrational numbers
3. Organization of mathematics, especially
geometry, into an ordred list of theorems
proved in a “logical” manner
4. Exact calculation of the areas and volumes of
several geometric objects (with proofs)
5. Essentially discovered the modern
mathematical concept of “limits”
The most important contributors
1.
2.
3.
4.
5.
6.
7.
Thales of Miletus (~625-547 B.C.)
Pythagoras of Samos(~580-500 B.C.)
Theaetetus of Athens(~415-369 B.C.)
Eudoxus of Cnidos (~408-355 B.C.)
Euclid of Alexandria (~300 B. C.)
Archimedes of Syracuse (~287-212 B.C.)
Appolonius of Perga (~262 B.C.-190 B.C.)
The Pythagorean school
1. Pythagoras (~580-500 B.C.) formed a school at
Crotona when he was about 50 years old.
2. The aims of the school were political,
philosophical and religious.
3. The community had the character of a secret
society with initiations, rites and prohibitions.
4. Pupils concentrated on four subjects of study:
arithmetic (in the sense of number theory),
music, geometry , and astronomy.
5. The school continued to exist for several
centuries after the death of Pythagoras.
Commensurable quantities and rational
numbers
• The Pythagoreans (the followers of Pythagoras)
believed that any two line segments are
commensurable, i.e there is a third segment, perhaps
very small, that could be marked off a whole number of
times on each of the given segments.
• In symbols, this says that given line segments of
lengths r and s, there is a third line segment of length t
such that 𝑟 = 𝑎𝑡 and 𝑠 = 𝑏𝑡 for some positive integers
a and b. It follows that 𝑟 𝑠 = 𝑎 𝑏 . In other words the
ratio of the lengths of any two line segments is a
rational number, i.e. a ratio of two integers.
The discovery of irrational numbers
• The Pythagoreans proved (most likely in the
5th century B.C.) that the diagonal of a square
is incommensurable with any of its sides. It
follows that 2is an irrationalnumber, i.e. it
cannot be written as a ratio of two integers.
s
d
s
𝑑=𝑠 2
The first crisis in mathematics (~5th
century B.C.) and its resolution
1. The discovery of irrational numbers was a shock
to the Pythagoreans.
2. They had used their wrong belief that any two
line segments are commensurable in some of
their proofs. Thus, a new approach was needed
to salvage many of their theorems.
3. Around 370. B.C., Eudoxus (~408-355 B.C.)
devised a theory of proportions which resolved
the crisis by making it possible to give correct
proofs of the aforementioned theorems.
Aristotle (384-322 B.C.)
1. Student of Plato
2. Tutor of Alexander the great
3. Contributed to logic, philosophy, metaphysics,
mathematics, physics, biology, ethics, politics,
agriculture, medicine and theatre
4. Wrote more than 200 treatises of which only 31
survive
5. Systematized the science of logic which was
already in use by the Greeks
Aristotle’s logic
1. The heart of Aristotle’s logic is the syllogism.
2. An example of a syllogism is:
“All men are mortal. Socrates is a man.
Therefore Socrates is mortal.”
Note that as long as the premises are true, the
conclusion must be true.
3. The syllogistic form of logical argumentation
dominated logic for more than 2000 years.
The Elements of Euclid (~300 B.C.)
1. A compilation of the most important mathematical
facts available at that time
2. Organized into 13 parts or books
3. Unified a collection of isolated discoveries into a
single deductive system based on a set of initial
definitions, postulates and common notions
4. Was the standard introduction to geometry until the
early 20th century
5. Modern introductions to geometry differ from the
Elements in logical order and proofs of theorems but
little in actual content.
Some definitions from the Elements
1. A point is that which has no parts.
2. A line is a being without breadth.
3. Parallel lines are straight lines which, being
in the same plane and being produced
indefinitely in both directions, do not meet
one another in either directions.
The Postulates (axioms) of the
Elements
1. A straight line can be drawn from any point to any
other point.
2. A finite straight line can be produced continuously in
a line.
3. A circle may be described with any center and
distance.
4. All right angles are equal to one another.
5. If a straight line falling on two straight lines makes the
interior angles on the same side less than two right
angles, then the two straight lines, if produced
indefinitely meet on that side on which are the angles
less than two right angles.
Flaws in the Elements
1. Some definitions are not satisfactory.
2. Some terms are used but not defined.
3. The postulates are incomplete, i.e. certain
tacit assumptions were used in the deductions
(proofs) which should have been included as
postulates or derived from them as
propositions (theorems).
Fixing the flaws in the Elements
1. The flaws could be fixed by introducing some
undefined terms (from which all other
definitions must be made) and a complete set of
postulates (or axioms). Such a collection would
be called an axiom system for Euclidean
geometry.
2. From the late 19th century onwards, many
mathematicians attempted to give a complete
statement of the postulates needed for proving
all the familiar theorems of Euclidean geometry.
3. These attempts are regarded as satisfactory.
The most popular of these attempts
1. David Hilbert (1899): 21 postulates , 6
undefined terms, namely point, straight line,
plane, on, congruent, and between
2. George David Birkhoff (1932): 4 postulates ,
4 undefined terms, namely: point, straight
line, distance, and angle
Note: Birkhoff ‘s postulates related geometry
to real numbers.
The basic underlying philosophy
• The only assumptions (about the various
geometric objects) which may be used in the
proof of a theorem are the postulates or the
theorems which have already been proved. No,
other “tacit assumption” or “intuitively obvious
fact” may be used.
Hilbert expressed this by saying “ One must
be able to say at all times – instead of points,
straight lines and planes – tables, chairs and
mugs.” ( Hilbert by Constance Reid)
Symbolic or Mathematical Logic
1. Although the Greeks considerably developed
the science of logic and Aristotle
systematized the material, the early work was
all carried out with the use of ordinary
language.
2. Modern mathematicians have found it
necessary to have a symbolic language in
order to further develop the science of logic.
A dose of Modern Symbolic Logic :
Propositional Logic
• Propositional logic deals with propositions.
• A proposition is a declarative sentence (i. e.
a sentence that declares a fact) that is either
true or false but not both.
• A propositional variable is a variable that
represents propositions.
• Propositional variables are usually denoted by
letters such as p, q, r, s etc.
Logical operators
¬ stands for “Not”, ˅ stands for “or”, ˄ stands
for “and”. These are defined using truth tables
as follows:
p
T
F
¬𝑝
F
T
p
T
T
F
F
q
T
F
T
F
p˅q
T
T
T
F
p
T
T
F
F
q
T
F
T
F
p˄q
T
F
F
F
More logical operators: Implications
• → stands for “implies” , ↔ stands for “if and
only if”
• 𝑝 → 𝑞 is also read as “if p then q”
p
T
T
F
F
q
T
F
T
F
p → q
T
F
T
T
p
T
T
F
F
q
T
F
T
F
p↔q
T
F
F
T
The NOR operator ↓
This is defined as follows:
p
q
p↓q
T
T
F
T
F
F
F
T
F
F
F
T
Note that 𝑝 ↓ 𝑞 is the same as ¬(𝑝 ˅ 𝑞).
Some things of interest in propositional
logic
1. To find the truth table of a compound
proposition (i. e. a sentance made up using
propositional variables and logical operators)
2. Given a truth table involving any number of
propositional variables, to find a corresponding
compound proposition
3. To study and develop relationships between
various logical operators
4. To study the notion of functional completeness
Functional Completeness
1. A collection of logical operators is said to be
functionally complete provided that every
compound proposition can be expressed as a
compound proposition involving only these
logical operators.
2. It is not hard to show that the collection
{¬, ˄} is functionally complete as is the
collection {¬, ˅}.
3. The collection ↓ is fuctionally complete.
The Liar Paradox
1. The earliest attribution is to Eubulides of Miletus
(~ 4th century B.C., contemporary of Aristotle)
who said “A man says that he is lying. Is what he
says true or false?”.
2. Alternate version: “This sentence is false”
3. In symbols : L is the sentence “L is false”. This
means that L is true if and only if L is false.
4. In propositional logic, such sentences are not
considered.
The cannibals puzzle
• An explorer visiting an island is captured by a
group of cannibals who reside on the island.
There are two types of cannibals: those who
always tell the truth and those who always lie.
The cannibals will barbecue the explorer
unless he can determine whether a particular
cannibal always lies or always tells the truth.
What one question can he ask the cannibal to
save himself?
Solution to the cannibals puzzle
Ask the cannibal any question whose correct
answer is “yes”. A truthful cannibal will answer
“yes” and a liar will answer “no”. One possible
question is “Are you a resident of this island?”.
Another possibility is “ Are you a cannibal?”.
The “villager and ruins” puzzle
Each inhabitant of a remote village always tells
the truth or always lies. A villager will only give
a “Yes” or a “No” response to a question a
tourist asks. Suppose you are a tourist visiting
this area and come to a fork in the road. One
branch leads to the ruins you want to visit; the
other branch leads deep into the jungle. A
villager is standing at the fork in the road. What
one question can you ask the villager to
determine which branch to take?
V
A propositional logic approach to the
“villager and ruins” puzzle
• r: The road on the right
leads to the ruins you
want to visit
• v: The villager standing
at the fork is truthful
• S: Statement to be
constructed
• A: Answer by villager to
the question “ Is S true”
v r
S
A
T T
T F
F T
T Yes
F No
F Yes
F F
T No
• One possible S is
¬𝑣 ˅ 𝑟 ˄ (𝑣 ˅ ¬𝑟)
Another solution to the villager and
ruins puzzle
If I were to ask you if the road on the
right leads to the the ruins, would
you answer “yes”?
Raymond Smullyan (1919-)
Raymond Smullyan (brief biography)
• Dropped out of school
• After moving from one university to another, he
was given an undergraduate degree in
mathematics by the University of Chicago in 1955
• Obtained a Ph. D. in logic in 1959 at Princeton
• Taught at Dartmouth College, Princeton
University, Yeshiva University, the City
University of New York, and Indiana University
• Wrote several books on logic, puzzles and chess
How he got his undergraduate degree
• Took the College Board Exams and was accepted by
Pacific University in Oregon
• Later studied at Reed College, University of Wisconsin
and the University of Chicago
• Got a teaching position at Dartmouth College based on
someone’s recommendation and a paper he had written
• He was a little short of the number of credits required
to graduate from the University of Chicago
• The problem was solved by giving him credit for a
calculus course he was teaching at Dartmouth but had
never taken
His books about puzzles
•
•
•
•
•
•
•
•
•
•
•
•
•
(1978) What Is the Name of This Book? The Riddle of Dracula and Other Logical
Puzzles - knights, knaves, and other logic puzzles
(1979) The Chess Mysteries of Sherlock Holmes - introducing retrograde analysis in
the game of chess.
(1981) The Chess Mysteries of the Arabian Knights - second book on retrograde
analysis chess problems.
(1982) The Lady or the Tiger? - ladies, tigers, and more logic puzzles
(1982) Alice in Puzzle-Land
(1985) To Mock a Mockingbird - puzzles based on combinatory logic
(1987) Forever Undecided - puzzles based on undecidability in formal systems
(1992) Satan, Cantor and Infinity
(1997) The Riddle of Scheherazade
(2007) The Magic Garden of George B. And Other Logic Puzzles, Polimetrica
(Monza/Italy)
(2009) Logical Labyrinths, A K Peters
(2010) King Arthur in Search of his Dog
(2013) The Godelian Puzzle Book: Puzzles, Paradoxes and Proofs
A puzzle from Raymond Smullyan
The Politician Puzzle.
A certain convention numbered 100 politicians. Each
politician was either crooked or honest. We are given
the following two facts:
• At least one of the politicians was honest.
• Given any two of the politicians, at least one of the
two was crooked.
Can it be determined from these two facts how many of
the politicians were honest and how many of them were
crooked?
Solution to the Politician Puzzle
Let A be an honest politician. Let B be any
other politician. Since at least one of A and B
is crooked, it follows that B is crooked. This
applies to any politician other than A.
Therefore, there is one honest politician and
the other 99 are crooked.
One last puzzle
Consider the following wrong equation:
5  5  5  550
How can you correct this equation by adding a
single straight line segment? The following is
not allowed:
5  5  5  550
Some contributors to the development
of symbolic logic
1.
2.
3.
4.
5.
6.
7.
8.
9.
Gottfried Leibniz (1646-1716)
George Boole (1815-1864)
Augustus De Morgan (1806-1871)
Charles Sanders Pierce (1839-1914)
Ernst Schröder (1841-1902)
Gottlob Frege (1848-1925)
Giuseppe Peano (1858-1932)
Bertrand Russell (1872-1970)
Alfred North Whitehead (1861-1947)
The second crisis in mathematics: The
foundations of calculus
1. Calculus was developed in the 17th century
and expanded considerably in the 18th
century.
2. The mathematicians involved in this process
failed to consider sufficiently the solidity of
the base upon which the subject was founded.
3. With the passage of time, contradictions arose
and it became necessary to properly deal with
the foundations of calculus.
Resolution of the second crisis
1. Augustin-Louis Cauchy (1789-1857) , Karl
Weierstrass (1815-1897) and his followers
resolved the crisis by successfully developing
a theory of limits.
2. They showed that all of existing calculus
(and related areas) can be logically derived
from a postulate set (or axiom system)
characterizing the real number system.
Further work on foundations (Late 19th
century)
1. It was shown that Euclidean geometry can be
based upon the real number system, (i. e. the
undefined terms of Euclidean geometry can be
defined in terms of real numbers and then the
postulates of Euclidean geometry can be proven
using the postulates for the real number system).
2. It was shown that the real number system can be
based upon a postulate set for the natural number
system.
3. In order to carry out these programs, basic
set theory is required.
The third crisis in mathematics:
Paradoxes in set theory
1. Georg Cantor (1845-1918) systematically
developed modern set theory.
2. It was hoped that all of mathematics can be
made to rest upon set theory as a foundation.
3. The discovery of several paradoxes in set
theory in the late 19th and early 20th century
was a shock.
Bertrand Russell (1872-1970)
Russell’s paradox
• Bertrand Russell (1872-1970) discovered the
following paradox:
Consider the set of all sets which do not
belong to themselves (such a set was allowed
in Cantorian set theory). Does this set belong
to itself? Either possibility leads to a
contradiction.
In symbols, let N  set of all sets X such that X  X .
Thus, for any set X , we have X  N  X  X .
Therefore, N  N  N  N .
Axiomatic set theory: A way to avoid
the paradoxes of set theory
1. In the first half of the 20th century, many
different axiom systems were developed for
set theory which avoided all known
paradoxes.
2. Unfortunately, these systems do not provide
any guarantee that no new paradoxes will
arise.
David Hilbert (1862-1943)
Hilbert’s dream
1. David Hilbert (1862-1943) developed “proof
theory” or “metamathematics” in which logical
reasoning itself was put into an axiomatic
framework.
2. This led to the notion of a formal theory or
system (a set of postulates together with a set of
rules of inference, i.e. a logic).
3. His “dream” was to prove that any formal
system that was rich enough to cover all of
mathematics was consistent, i.e. did not produce
any contradictions.
Kurt Gödel (1906-1978)
Gödel’s Incompleteness Theorems
(1931): End of Hilbert’s dream
1. First incompleteness theorem
Any consistent formal system F within which a
certain amount of elementary arithmetic can be
carried out is incomplete; i.e., there are
statements of the language of F which can
neither be proved nor disproved in F.
2. Second incompleteness theorem
For any consistent system F within which a
certain amount of elementary arithmetic can be
carried out, the consistency of F cannot be
proved in F itself.
A statement of F. De Sua
Suppose we loosely define a religion as any
discipline whose foundations rest on an
element of faith, irrespective of any element of
reason which may be present. Quantum
mechanics for example would be a religion
under this definition. But mathematics would
hold the unique position of being the only
branch of theology possessing a rigorous
demonstration of the fact that it should be so
classified.
Non-Aristotelian logics
1.
2.
3.
4.
Many valued logic
Intuitionist logic
Infinite valued logic
Fuzzy logic
The intuitionist school
1.
2.
3.
4.
5.
6.
Started in early 20th century by L. E. J. Brouwer although some of
the main ideas were mentioned earlier by Kronecker (1823-1891).
Many eminent mathematicians have joined this school.
The main idea is that any mathematical object must be built in a
purely constructive manner, employing a finite number of steps or
operations from the natural numbers.
They deny the universal applicability within mathematics of the
“fact” that any clear mathematical statement must be either true or
false.
They have succeeded in building large parts of present day
mathematics according to their principles.
It is generally conceded (not proven) that their methods do not lead
to paradoxes.
Conclusion
1. Mathematical Logic and related subjects such
as Set Theory have continued to be subjects
of intense research and controversy until
today.
2. The Game of Logic that began with the
Greeks is still thriving.
Solution to “last” puzzle
5+5+5 = 550
5+5+5 = 550
Thank You